Chapter 2
SYSTEMS OF EQUATIONS
In the previous chapter the reader was acquainted with equations having one
unknown function. However, almost all partial differential equations can be
reduced to a system of quasilinear equations.
The first section of this chapter is devoted to explaining classical knowledge related to systems of quasilinear partial differential equations. The main
definitions of quasilinear systems, as well as the notions of characteristics and
relations along characteristics are considered in this section.
Systems written in Riemann invariants play an important role in continuum mechanics. In particular, homogeneous systems have solutions, called
Riemann waves, where only one Riemann invariant is changed. The wellknown problem of the decay of arbitrary discontinuity is solved in terms of
Riemann waves. Chapter 2 also contains an application of Riemann waves for
describing one-dimensional motion of an elastic-plastic material.
Another method playing a very important role in gas dynamics is the hodograph method. For some problems this method allows linearization. If a hodograph is degenerate, such solutions form a class of solutions called solutions
with degenerate hodograph. This class of solutions is considered in the next
chapter. Here it is worth mentioning that solutions with degenerate hodograph
have a group-invariant nature: they are partially invariant solutions. Another
class of solutions which also has group-invariant nature is the class of selfsimilar solutions. This class of invariant solutions, very well-known in continuum mechanics, is based on the analysis of dimensions of studied quantities.
The approach used in the book for introducing self-similar solutions is related
to admitted scale groups. This way of studying self-similar solutions can also
be considered as an introduction to group analysis method. The theory of selfsimilar solutions is followed by applications to one-dimensional gas dynamics,
in particular to the problem of an intense gas explosion.
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Among the approaches using a simple representation of the dependent variables through the independent variables are travelling waves and solutions with
linear dependence of velocity with respect to spatial (all or part) independent
variables.
The chapter ends with a general study of completely integrable systems.
Basic definitions
Most mathematical problems in science are described by systems of partial
differential equations. The vast majority of these problems are reduced to problems involving the study of systems of quasilinear partial differential equations.
In this section one can find sufficient notations for further reading1 related with
these types of systems.
Let x = (xl, x2, . . . ,x,) E Rn be the independent variables and u =
( u l , 242, . . . , u,), E Rm be the dependent functions.
Definition 2.1. A system of first order partial differential equations of the
form
is called a system of quasilinear partial differential equations.
Any system of quasilinear partial differential equations can be written in the
matrix form
where A, are rectangle N x m matrices with the entries a$ (i is the row number,
/3 is the column number), and the vector-columns
System (2.1) or (2.2) is called a determined system if N = m , and overdetermined if N > m. If the vector f and the matrices A,, (a = 1, 2, ..., n )
do not depend on the independent variables x = (xl, x2, . . . ,x,), then the system is called an autonomous system. If the vector f = 0, then it is called a
homogeneous system.
The matrix
qu or more detail study of properties of systems of quasilinear partial differential equations one can read, for
example, [147].
Systems of equations
associated with system (2.2) is called the characteristic matrix. Here
(<1,<2, .-.,<n) E Rn.
<
=
<
Definition 2.2. A vector is called a normal characteristic vector of system
(2.2)at the point ( x , u ) , i f
(2.3)
Equation (2.3)is called the characteristic equation.
For any normal characteristic vector there exists a left eigenvector 1 (t)=
( 1 1 0 ) l2(<),
~
. . . , l n z ( t )of
) the matrix A ( t )
<
Let
r] E
Rn be a unit fixed vector, then any vector
< can be decomposed
where a is a vector orthogonal to r ] .
Definition 2.3. System (2.2) is called a hyperbolic system at a point ( x , u ) ,
if there exists a vector r ] , such that for any vector a , which is orthogonal to r ] ,
the characteristic equation
+
det (A(zr] a ) ) = 0
has m real roots with respect to z and the set of eigenvectors l k ,
( k = 1,2, . . . , m ) corresponding to these roots, composes a basis in Rm. A
system is called a hyperbolic system in a domain D , i f it is hyperbolic at any
point ( x , u ) E D (the vector r] can depend on ( x , u)). A direction defined by
the vector r] is called a hyperbolic direction.
Let a solution u = @ ( x )of system (2.2) be given. The point ( x , u ) =
( x, @ ( x ) )is defined by the point x .
Definition 2.4. A sugace l- c Rn ( x )such that a tangent hyperplane at any
x E l- has a characteristic direction is called a characteristic surface of system
(2.2)on this solution, or simply a characteristic.
The problem of finding a characteristic given by the equation h ( x ) = 0 is
reduced to the problem of solving the equation
det ( A ( V h ) ) = 0.
This equation is a first order partial differential equation. The characteristics of
this equation are called bicharacteristics of system (2.2).
Remark 2.1. If one of the matrices Ai, (i = 1,2, ...,n ) is nonsingular,
then without loss of generality one can assume that A l is the unit matrix.
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Remark 2.2. The concept of characteristic surfaces plays an important
role in the process of constructing solutions. Their allocations give information about the qualitative behavior of the solution, and about the correctness
of initial and boundary value problems. Moreover, they play a crucial role in
defining the so called "simple" solutions. Sewing together simple solutions
one can find more "complicated" solutions. The classical example of this construction is the solution of the decay of an arbitrary discontinuity problem in
gas dynamics.
Riemann invariants
In this section the Riemann invariants for a hyperbolic system of quasilinear
partial differential equations with two independent variables2
are considered. The direction of hyperbolicity of system (2.5) is assumed
q = ( 1 , 0). Hence, the matrix A has m real eigenvalues h k ( x ,u ) with the
left eigenvectors l k ( x ,u ) , (k = 1 , . . . , m ) of the matrix A. The eigenvectors
1 , u ) (k = 1 , . . . m ) form a basis for R"'. Hence, the matrix L = (1;)
comprising the coordinates 1: ( x, u ) of the vectors 1 ( x, u ) is nonsingular. If
the characteristic, corresponding to the eigenvalue h k ( x ,u ) of system (2.5) is
defined by the equation hk = x2 -q5k ( x l )= 0 , then the function q5k ( x l )satisfies
the equation dq5k/dx1 = hk.
Multiplying system (2.5) by the left eigenvectors l k ( x ,u ) , one obtains:
.
= l k f , ( k = 1,2, . . . , m).
lk(x,u)
Let each of the differential forms
mk(x,u , du) = I,k du,,
k = l , 2 , . . . , nl
with fixed x l , x2 have the integrating factor3 p k ( x , u ) , i. e.,
Hence, after multiplying equations (2.6) by pk, they take on the form
notation of Riemann invariants in the case of more than two independent variables is given in the next
Chapter.
3 ~ the
n case m = 2 the integrating factor always exists. This is not so form > 2.
2~
Systems of equations
+
+
where gk = p k fk
rhxl AkriX2,D,, is the total derivative operator with
respect to xi, and the value rLxi is the partial derivative of the function rk with
respect to xi with fixed values of the dependent variables ul, ua, . . . , urn. Since
the Jacobian
changing the dependent variables (ul, . . . , u,) into the new dependent functions (rl, . . . , r,), one reduces system (2.5) to the system of quasilinear partial
differential equations
ark
ark
= gk, (k = 1 , 2, . . . , m).
8x1
ax2
The values rk are called Riemann invariants and system (2.7) is a system written in invariants. If system (2.5) is homogeneous and autonomous (A =
A(u), f = 0), then the system written in invariants (2.7) is also homogeneous.
If gk = 0 in (2.7), then the Riemann invariant rk is constant along the characteristic dx
;t;. = Ak.
Let a system written in Riemann invariants (2.7) be homogeneous gi =
0, (i = 1 , 2 , . . . , m). Assume that all Riemann invariants except one, for
example rk, are constant, i. e.,
- +Ak(x,u)-
ri = c i , (1 5 i 5 m, i # k).
The Riemann invariant rk satisfies the equation
A
where hk(x,rk) = hk(x,c l , . . . , ck-1, rk, ck+l, . . . , c,).
called a simple wave or a Riemann wave.
2.1
Such a solution is
The problem of stretching an elastic-plastic bar
The equations, governing a one-dimensional time dependent motion of an
elastic-plastic material with the state equation a = f (E), f' = @ 2 ( ~>) 0, are
Hence, the Riemann invariants and characteristic eigenvalues are
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
where F ( 8 ) =
(2.8) becomes
1;:@ (q)dq. The original system of partial differential equations
2 + hl(r1, 1-212 = 0,
ar2
ar
+ 3L2(r1,72)3$
= 0,
The dependent variables are recovered through the Riemann invariants
Let us pose the problem of stretching a semi-infinite bar. Assume that at the
initial time t = 0 an elastic-plastic half-infinitely long bar is in the unperturbed
state:
v(x, 0) = 0, E ( X 0)
, = r,,, x 3 0.
The end of the bar x = 0 at the time t = 0 starts stretching with the velocity
It is assumed that u(0) = 0 and u f ( t ) 2 0. The problem is to find the loading
wave propagating in the bar.
The solution of this problem can be constructed with the help of Riemann
waves.
Since the Riemann invariants are constant along their own characteristics,
the solution in the domain, joining to the initial data V1 = ( ( x ,t ) I 0 5 t 5
oo, 0 5 x 5 t @ ( r 0 ) }is, constant: v = 0, r = ro. Because the Riemann
invariant r l is constant along the characteristics d x l d t = -@ (which cross the
characteristic curve x = t @( r 0 ) )and have a constant value on the characteristic
x = t@(cO),
this invariant is also constant in the domain V2,joining to V1.
This means, that in this domain one obtains the Riemann wave: r l = v
F ( r ) = F (ro).In this Riemann wave the other Riemann invariant 7 2 constant
along the characteristics d x l d t = @, which are straight lines. Hence, the
solution in the domain V2 is defined by the values v = -u(t) and F ( E ) =
F ( E ~ ) u ( t ) at the point (xo(t),t ) , where xo(t) = - u ( s ) ds. The relation
u f ( t ) 3 0 provides the condition that characteristics
= @ intersect in the
domain V2. If the condition u f ( t ) > 0 is broken, this leads to the formation
of a gradient catastrophe. The relation u (0) = 0 gives a nonsingular Riemann
wave. If u (0) > 0, then the part of the domain V2is occupied by the rarefaction
),
Riemann wave. This part is bounded by the characteristic x = t @( E ~ where
F ( r l ) = F ( r o ) u(0). The deformation r in this domain is defined by the
= @ issue from the origin ( x , t ) =
equation @ ( E )= The characteristics
+
+
2
+
r.
(090).
Hodograph method
The basic idea of the hodograph method consists of interchanging the role
of the dependent and independent variables. For some classes of equations this
Systems of equations
method reduces them to a system of linear partial differential equations. The
essence of the hodograph method is described by the system of the equations,
governing two-dimensional irrotational isentropic flows of a gas (v = 0,1):
where (u,v) is the velocity, c is the sound speed, which is expressed through
the value q 2 = u 2 v2. The plane R ~ ( u ,v) is called a hodograph plane, and
(u, v) are the hodograph variables.
Assume that the ~acobian?A = - # 0. Choosing (u,v) as the new
independent variables, one can find
+
x = x(u, v), y = y(u, v).
Differentiating these relations with respect to x and y , one obtains
1 =x,ux +x,v,, o = y,u, +yvvs,
0 = xuuy + xuvy, 1 = yuuy + yvvy.
Since A# 0,one finds
The two-dimensional gas dynamics equations (2.10)may then be written
The first equation of (2.12) leads to the existence of a potential: a function
4 = 4(u,v) such that
x =@u, Y =4v.
The second equation of (2.12)becomes the equation for the function 4
Equation (2.13)assumes a specially simple form in the case v = 0 in the
polar coordinates (u = q cos 8 ,v = q sin 8):
where M = q / c is the Mach number. The most important property of equation
(2.14)consists of its linearity. Thus the hodograph transformation can simplify
4~anishingof the Jacobian A defines a class of solutions which are called solutions with degenerate hodograph. A detailed study of these solutions is given in the next Chapter.
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
the original system of equations. It should be also noted that equation (2.14)
allows finding solutions with separated variables q and 8. In fact, substituting
the representation of the solution
into (2.14), one obtains
Alongside the complete interchange of the dependent and independent variables, sometimes it is convenient to make only a partial change of them. For
example, for the equations of a stationary boundary layer
applying the transformation from the independent variables ( x , y) to the
Prandtl-Mises variables: 6 = x , q = I)( x , y), one obtains the equations
where u = l / ( a y / dy ) . Introducing the variable 22 = lJ2 - u2, one finds the
equation of a boundary layer in the Mises form
4. Self-similar solutions
4.1 Definitions and basic properties
One of the modelling stages of a problem in continuum mechanics is the
dimensional analysis of the quantities of the variables involved. This analysis
also allows forming representations of solutions, which are called self-similar
solutions. One example of a self-similar solution was presented in the first
chapter. Here the main definitions and properties of self-similar solutions are
given. Since the basis for dimensional analysis is a scale group, the given
approach is based on the concept of an admitted scale group.
Let ( x l ,. . . , x,?) and ( u l , . . . , u,) be the independent and dependent variables.
Definition 2.5. A transformation ha : Rnf" + R ~ of+the~form
Systems of equations
is called a scale group H' of transformations of the space R"+m(x,u ) . The
variables a, (a = 1 , . . . , r ) are called its parameters.
It is natural to require
Otherwise, introducing new scale parameters, it is possible to reduce the number r .
Under action of the transformation (2.15)the first order derivatives are transformed according to the formulae
Similar formulae are valid for higher order derivatives.
Definition 2.6. The group of transformations, consisting of the transformations of the independent, dependent variables (2.15)and the derivatives (2.17),
is called a prolonged group of H' .
For any function F : R"+m + R the total derivative with respect to the
parameter aj is
Definition 2.7. The linear differential operator
is called an infinitesimal operator of the group H r .
Definition 2.8. A function F : R n f m + R is called an invariant of the
group H r , iffor any transformation ha E H r :
F ( x f ,u') = F ( x , u ) .
Theorem 2.1. A function F : Rn+m + R is an invariant of the group H'
if and only i f
(2.20)
cjaF = 0 , ( j = 1 , . . . , r ) .
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Proof.
If the function F ( x, u ) is invariant, then F (x', u') = F ( x, u ). Differentiating it with respect to the parameter a j , ( j = 1 , . . . , r), and setting the parameters ai = 1 , (i = 1 , 2, ...,r ) , one obtains (2.20). Conversely, if (2.20) is valid,
then by virtue of (2.18) one has d F ( x f ,u f ) / a a j = 0 , ( j = 1 , . . . , r ) . This
means that F (x', u') does not depend on the parameters aj , ( j = 1 , . . . , r ) .
Since for aj = 1 , (i = 1,2, ..., r ) its value is equal to F ( x , u ) , hence,
F (x', u') = F ( x , u). Therefore F is an invariant of the group Hr .a
Theorem 2.2. For a scale group Hr of the space Rn+"(x, u ) with the
condition r < n m there exist n m - r independent invariants. They are
the monomials
+
+
Proof.
By virtue of the criterion, the monomial
is an invariant of the group Hr , if and only if
+
Because of (2.16), the system of linear algebraic equations (2.22) with n m
unknown 8, , ay has n +m - r linearly independent solutions. Let the solutions
be
k
(of, ..., OkH , c rkl , ..., mm),
(k = 1 , ..., m + n - 7 ) .
(2.23)
Then the monomials (2.21) with exponents (2.23) are also independent. By
virtue of the exponent representation of the monomials (2.21), it is enough to
prove the independence of their exponent.
Assume that there exist constants ck, (k = 1 , . . . , m n - r ) , at least one
. . . c;+,-, # O), and for which one
of them is not equal to zero (c: C;
has the equality
+ + +
n+m-r
This is only possible if
+
Systems of equations
But by virtue of the linear independence of (2.23), one obtains ck = 0 , (k =
1, . . . , m n - r ) . This contradicts to the assumption c: c; . . . ci+,-, #
0..
+
+ + +
Definition 2.9. A manifold assigned by the equations dk( x , u ) = 0 , (k =
1,2, . . . , 1), is called an invariant manifold of the group H r , if
The concept of a scale group is closely related with the dimensional analysis
theory of physical quantities [130].
Each physical quantity 4 is characterized by a unit of its measurement E
and a numerical value 141, hence, 4 = I@ I E . Let E a , (a = 1, . . . , r ) be
some independent units of measurements that E is expressed through them
E =
E?. The value E is called the dimension of the physical quantity
4 in the terms of the units E, and it is denoted [@I.
If one changes the scales of the units Ei into the new units E f
nL=,
Ei = aiEi, ( i = 1 , . . . , r ) ,
the numerical value of the physical quantity @ in the new units is
This means that 141' = 141( n L = l (a,)ha), i.e., the change of the numerical
values of the physical quantity 4 is similar with the scale group H'. When
constructing exact solutions by the dimensional analysis theory one can use the
theory of invariant solutions with respect to the scale group H'. This theory is
explained next.
Definition 2.10. A scale group H r is said to be an admitted by a system
of partial differential equations if the manifold assigned by this system is invariant with respect to the prolonged group H r .
Nonsingular invariant solutions are constructed as follows. First, one finds
, = 1 , . . . , m n - r ) . They should be
all of the independent invariants J ~ (k
such that it is possible to solve m of them (for example, J k , ( k = 1, . . . , m ) )
with respect to all dependent variables. A sufficient condition for this is the
inequality
+
(n
m
m
a ( J 1 , .. . , J n l )
=
Jk/
u,) det
a ( u l , .. . ,u m )
k=l
a=l
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Without loss of generality one can let of. = Jij (i, j = 1, . . . , m). The remaining invariants J ~ + (k
~ =
, 1, . . . , n - r ) can be chosen depending only of the
independent variables (the case where r = n is also possible). Hence,
+
After obtaining the independent invariants J k , (k = 1, . . . , m n - r ) , one
supposes the dependence of the first invariants J k , (k = 1, . . . , m) of the
remaining, i.e.,
J k = (ok(Jm+l,. . . , ~nz+n-r), ( k = 1,..., m).
Since the invariants J k , (k = 1, . . . , m) can be solved with respect to all
dependent variables u' , (i = 1, ...,m), defining the dependent variables from
the last equations, one obtains a representation of the invariant solution. Substituting the representation of the functions uZ, (i = 1, ...,m) into the initial
system of partial differential equations, one obtains the system of equations for
the unknown functions ( ~ k ,(k = 1, . . . , m). This system involves a smaller
number of independent variables.
Definition 2.11. An invariant solution of an admitted scale group H ' is
called a selj-similar solution5.
Let us apply the theory to the nonlinear diffusion equation6
The dependent and independent variables are scaled as follows
r = aU2r, t = aU3t.
A
c=
Equation (2.24) in the new variables becomes
If equation (2.24) is invariant with respect to scaling, then it is necessary that
nal
- 2a2
+ a 3 = 0.
For a 3 # 0 the combinations J 1 = ~ t - ~ l I ~J32 ,= 1"t-a2/ff3are invariant with
respect to this scaling. Assuming
'.-ff1Iff3 = ~ ( ~ ~ - a 2 l f f 3 ) ,
one obtains the representation of an invariant (self-similar) solution of (1.56).
'with a group point of view such solutions are called self-similar solutions in narrow sense [130].
6~elf-similarsolutions of this equation were considered in the previous chapter.
Systems of equations
4.2
Self-similar solutions in an inviscid gas
The one-dimensional gas dynamics equations are considered to illustrate the
method of constructing self-similar solutions. Notice that the solution of the
problem of a strong explosion in a gas was found with the help of self-similar
solutions.
The system of equations describing a one-dimensional motion of a gas is
Here y is the exponent of the adiabatic curve, v characterizes a geometrical
structure of the problem: v = 0 for plane flows, v = 1 for cylindrical flows, and
v = 2 for spherical flows. Because the number of the independent variables
n = 2, by virtue of the condition n > r only two cases are possible: either
r=lorr=2.
Let r = 1, and define the one-parameter scale group H' admitted by system
(2.25) with the equations (a = al):
For the transformations that remain invariant the manifold, assigned by equations (2.25), one obtains
+
+
Further it is assumed that (h112 (h2)2 # 0. Since m n - r = 4, when
forming the independent invariants J~ = t8:x@ua:pai pal, (k = 1 , 2 , 3 , 4 ) it
is enough to find independent solutions of the linear equation
If h1 = 0, one can choose the following independent invariants
The self-similar solution has the representation7
u = x41 (t), p = xUq!Q(t), p = ~ ~ + ~ $ ~ ( t ) .
The system of equations (2.25) becomes
7 ~ u c solutions
h
are called solutions with a linear profile of velocity.
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
where @ = @-3/&.
The solution of this system of equations can be found in
terms of quadratures.
If h1 # 0, the invariants can be chosen as follows
l . self-similar solution in this case has
where a = -h2/h1, /? = - / ~ ~ / h The
the representation
where h = x t a . There is another equivalent representation
X
u = -U (A), p = X - ( ~ + ~ ) ~ - S (A),
R
p = x - ( k + 1 ) t - ( s + 2 ) ~ ( A ).
t
(2.26)
Substituting these expressions in equations (2.25), one obtains the system of
ordinary differential equations for the functions U , R , P
This system of equations is split into two equations of the form
dU
dR
A- d h = f i ( U , Z ) , A= ~f
dh
2(U, Z ) ,
and one differential equation
where Z = P I R . Notice that the last equation in the case where k = -3, s =
-(2 a ( v 3 ) ) has the integral
+
+
The values of the constant a and /? in each particular self-similar solution
are chosen by analysis of the initial parameters of the problem.
4.3
An intense explosion in a gas
The problem of a strong explosion in a gas is formulated as followss.
'~etailedanalysis of this problem can be found in [149]and references therein.
Systems of equations
At the moment t = 0 in an undisturbed gas (ul = 0) with the initial density
pl and zero pressure p l = 0 at the center of symmetry (x = 0) an explosion
occurs, i.e., a final energy Eo is instantly released. The areas of the disturbed
and undisturbed parts of a gas are separated by a shock wave. The gas dynamics
values in front of the wave pl, p l , ul and behind it p2, p2, u2 are related by
the conditions across the shock wave (the Hugoniot relations):
These relations express the conservation of mass, impulse and energy laws on
the shock wave x = xb(t). Here D = dxb/dt is the velocity of the shock
wave propagation. From the Hugoniot relations one can obtain the values of
the density, pressure and velocity behind the front of the wave
The disturbed part of the gas is located in the interval (0, xb). Because the
energy of the volume of the gas is equal to p 2 / 2 p/(y - I), according to
the conservation of energy
+
The motion of the gas after the explosion (t > 0) is defined by the dimension
of the parameters Eo, pl, x and t. They have the dimensions
[Eo] = M L " T - ~ , [pi] = M L - ~ , [XI = L , [t] = T.
Here M is the dimension of the mass, L is the dimension of the length and T
is the dimension of the time. In this problem there is only one dimensionless
variable parameter
1
2
A =(E/pl)-Oxt-O,
where E = hEo with some constant h. This constant is chosen to scale
the variable A. Assume that the disturbed part of the gas is governed by the
self-similar solution of the type (2.26). The front of the shock wave is xb =
1
h*t-e elpi) 0.
Without loss of generality one can let A, = 1. According
to the dependence of the variable h of t and x, in (2.26) one has to choose
a = -2/(v
3). Analyzing the dimension of the density [p] it also follows
that j3 = 0. Therefore when seeking the solution of the problem of an intense
explosion, one can try to find it in the class of self-similar solutions (2.26) with
s=O,k=-3:
+
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Notice that for these parameters there is the integral (2.28). Because the coorh , D = -axat-'.
dinate of the shock wave front is xb = t P ( ~ / ~ l ) then
This gives the initial data at h = 1 (on the front of the shock wave)
Equation (2.29) in the self-similar variables is reduced to the equation
This equation serves for finding the constant
Therefore, the solution of intense explosion can be found in the form of quadratures of (2.27).
Solutions with a linear profile of velocity
Among the approaches for obtaining classes of exact solutions in continuum mechanics there is the method where the velocity vector is required to be
linear with respect to the spatial independent variables, with respect to all or
their part of them. Such solutions were studied a long time ago by Dirichlet,
Dedekind and Riemann. One example of such a solution was obtained in the
previous section9. Assuming linearity of the velocity with respect to some independent variables, one usually obtains polynomial equations with respect to
them. Splitting these equations leads to an overdetermined system of partial
differential equations. The main problem in these studies is the compatibility
problem for the overdetermined system of equations.
Let us consider the equations describing the isentropic motion of a polytropic gas. For the sake of simplicity10 we consider the two-dimensional case
We will assume that the velocity vector has the representation
The last two equations of (2.30) define the derivatives
9~pplications
of this method to the Navier-Stokes equations can be found in [159]
'O~imilar,but more cumbersome, one can obtain solutions in the three-dimensional case.
Systems of equations
where
The condition
a2o/axlax2 = a2e/ax2axl
gives a12 = a21. Integrating (2.32) with respect to xl and x2, one finds
where @ ( t ) is an arbitrary function of integration. Substituting the expression
for 8 and the velocity in the first equation of (2.30), one obtains the squared
form with respect to xl and x2
with the coefficients
Factors of the squared form (2.34)are independent of the independent variables
xl and x2. Hence, splitting equation (2.34), one has a system of five ordinary
differential equations for five functions: @ ( t ) and mij( t ) , ( i , j = 1,2).
More general class of solutions is obtained if one assumes linearity only
with respect to one independent variable.
6.
Travelling waves
The idea of a travelling wave was presented in the previous chapter. The
concept of travelling waves can be generalized for many independent variables
x E Rn and many dependent variables u E Rm. In this section the generalization for the multidimensional case is given.
Let L x be a linear form of the independent variables
The representation of a solution is assumed in the form
u( x )= v(Lx),
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
where the function v ( ( ) depends on one independent variable (. The value (
is called a phase of the wave. Fixing the phase ( = L x , one obtains the front
of the wave, where the values of the dependent variables are constant. Hence,
the front of the wave is a plane propagating in the space Rn. Usually one of the
independent variables plays the role of time t (for example, t = x,), the phase
of the wave is represented as ( = qy - D t , where q E R,-' is a unit vector,
y = ( x l ,x2, ..., x,-l), and qy is a scalar product.
Definition 2.12. A solution u ( x l ,x2, . . . , x,) is called an r-multiple travelling wave, if it has the representation
where L x be a vector, which coordinates are linear forms of the independent
variables
( L X )=~ Liaxa, (i = 1,2, ...,r, r < n ) .
The variables { = L x E R" are called pammeters of the wave.
Here the vector function v ( ( ) depends on a smaller number of the independent variables (. Fixing all but one of the components of the vector defines a
wave propagating in the subspace of Rn of the dimension1' n - (r - 1 ) . Equations for the function v ( c ) are obtained by substituting the representation of the
solution into the initial system of equations.
Notice that for an r-multiple travelling wave the rank of the Jacobi matrix
of the dependent variables with respect to the independent variables is less or
equal than r. This provides an idea for further generalization of the travelling
wave concept. The generalization is achieved by rejecting the requirement that
the parameters of the wave are linear forms. These solutions are called solutions with a degenerate hodograph, and they are studied in the next chapter.
Let us apply the method to two-dimensional flows of a fluid, described by
the Navier-Stokes equations
c
Ut
vt
+
+
24U,
UV,
+ + p, = + Uyy,
+ vvy + py = v,, + v y y ,
VUy
U,
+
Uxx
Uy
= 0.
Without loss of generality one can assume that the travelling wave type solution
has the representation
Substituting the representation of the solution into the Navier-Stokes equations,
one obtains
p' + ( u + av + D ) u f = (a2 l ) u f ' ,
(2.35)
up' (u av D ) v f = (a2 l ) v f ' ,
U'
av' = 0.
+ + +
+
+
+
"Without loss of generality it is assumed that the rank of the matrix L = ( L a p )is equal to k.
Systems of equations
Taking the linear combination of the first equation and the second equation
multiplied by the constant a , and using the third equation, one finds
Integrating the third equation of (2.35), one gets
where c is constant. The second equation of (2.35) becomes
Notice that by virtue of a Galilean transformation and a rotation one can assume
that c = 0 and a = 0.
7.
Completely integrable systems
One class of overdetermined systems, for which the problem of compatibility is solved, is the class of completely integrable systems. One particular
case of such a system was considered in the first chapter. Here the theory of
completely integrable systems is developed in the general case.
Definition 2.13. A system
is called a completely integrable i f it has a solution for any initial values a,, z,
in some open domain D.
Lemma 2.1. Any system of the type (2.36) is completely integrable if and
only if the equalities
are identically satisfied with respect to the variables ( a , z )
Prooof.
Let z = z ( a ) be a solution of the initial value problem
z(a,) = z,.
Calculating the derivatives
E
D.
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
at the point a,, one obtains
Since the initial values a,, z, are arbitrary, the first part of the lemma is proven.
For proving the second part of the lemma, the theorems of existence, uniqueness and continuity with respect to parameters of a solution of an initial value
problem are applied. Let e be an arbitrary vector in the closed unit ball B1( O ) ,
and pose the problem
avi
-(t,e)
at
= e U f ; ( a , + t e , v ( t , e ) ) , v i ( 0 ) = z ; , ( i = 1 , 2,,..., N ) . (2.38)
This problem has a unique solution v ( t , e ) , which is defined in the maximal interval t E ( 0 , re). Direct calculations show that the function u ( t , e ) = v ( h t , e )
is a solution of the problem
a
~
l
-(t,e)=heUf;(a,+hte,u(t,e)),
at
u i ( 0 ) = z b , ( i = 1 , 2, , . . . , N ) .
Because of the uniqueness of the solution for he E B1( O ) , the vector function
u ( t, e ) = v ( t, he). Hence, v ( t , he) is defined in the interval t E ( 0 , the),
and
Athe = re.
Set t, = inf, t,, and assume that t, = 0. This means that for any E > 0 there
exists a vector e E B1( 0 ) such that t, = 0 5 t, < E . Choosing E = l / k one
constructs a sequence of the vectors { e k }such that t,, + 0. Because the unit
ball B1( 0 ) is a compact, there exists a convergent subsequence { e k l }+ e, E
B1(0). For the vector e, the solution of the problem (2.38) is defined in the
interval (0, t,,), where tee> 0. Because of the continuity with respect to the
parameter e, there exists an interval ( 0 , t )and a neighborhood U,, of the vector
e, such that 0 < t < teeand for any e E U,, the solution v ( t , e ) is defined
+ 0. Hence,
in the interval ( 0 , t ) .This contradicts to the condition that teLl
t*> 0.
Set t = min(1, t,). The functions vi ( t , a - a,) are defined for any a E
B,(a,) and t E (0, 11. Let us prove this statement.
If t, 2 1, this follows from the definition of t,. In fact, in this case t = 1,
and for any a E B, (a,) the functions vi ( t , e ) with e = a - a, E B1( 0 ) are
defined in the interval ( 0 , t ) ,where t > t, > 1. If t, < 1, then t = t,. For
- a 0 ), with
any vector a E B,* (a,) such that a - a, # 0 the vector e = h-'(a
h = la - a, I-' , belongs to the ball B1(0). Notice that he c B, ( 0 ) c B1(0).
Since t h e = A-l t, > h-'t, = h-' t 2 1, then for any a E B, (a,) the values
vi ( 1 , a - a,) are defined. Thus, the functions v' ( t , a - a,) are defined at the
point t = 1 foranya E B,(a,).
Systems of equations
Let us show that the functions zi ( a ) = vi ( 1 , a - a,), Va E B, (a,) satisfy
the equations
+
For this purpose the functions S; ( t ) = t R; (a, t e ) ,where e = a -a,
are studied.
First one can obtain some useful identities. Notice that
zi(a,
E
B, ( 0 ) ,
+ t e ) = v i ( l ,t e ) = v i ( t , e ) .
Differentiating these equalities with respect to t , and using equations (2.38),
one has
azl
eB-(a,
asp
+ t e ) = eSfj(ao + te. z(ao + re)).
which at the point t = 1 become
Differentiating these relations with respect to the coordinate a', one obtains
a
-(a)
ad
f;(a, ; ( a ) )
+ (up - a!) -( a ) =
+ (aS - 4)
Because of the definition S; ( t ) = t R ; (a,
last equations are
teB
(&("
+ afj
~ ( 0 ) )r n ( a ,z
( a ) ) s ( a ,; ( a ) )
+ t e ) , at the point a = a, + te the
af
+ t e ) ) = -R;(a, + re) + teB$(a,
eS ( ~ r ( t+) t f J ( a , + t e ) ) %(a, + te).
+re)+
From another point of view, differentiating the relations tR;(a,
( t ) with respect to t , one obtains
s;.
(2.39)
+ re) =
EXACT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Using (2.39),these equations are reduced to the equations
Changing t R; with S; ( t ) ,one finds
dsj(t)
dt
=
aft
-eB~iL
+eBsY+ teB
azy
azy
J
(:i
b +fyJ
afj
Y
af;
af j
-- f Y fi
azy
Because of the relations (2.37), one finds that the functions S\(t) satisfy the
problem
By virtue of the uniqueness of the solution of this problem, one finds that
S:(t) = 0 or ~ ; ( a=) 0.
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